How do I limit particles to move only on certain axis?
I have made Extra Nodes for Animation Nodes (blender 2.79, blender 2.80) which allows to control particles very easily. Here is solution for your problem:

So, in this node-tree, using particle data node, first get the locations of all particles (unborn, alive, dying which is crucial step) then multiply the locations with vector (1, 0, 1) which makes y-axis locations zero or limit the particles in the XZ-axis but you can limit them in any axes. Now, use these new locations of particles and plug them into the locations input of the particles output node. Here is tutorial link how to install and use Extra Nodes: https://www.youtube.com/watch?v=hvzr8lu5Ef0